Triangle-Covered Graphs: Algorithms, Complexity, and Structure

2025-09-14

AI Summary

This paper studies the triangle covering problem on graphs: adding the minimum number of edges so that every vertex belongs to at least one triangle—yielding a “triangle-covered graph.” We introduce the core notion of a Δ-completion set and provide the first systematic formulation of this edge-modification problem. Theoretically, we prove NP-completeness and inapproximability within any constant factor; further, using probabilistic methods, we establish the phase transition threshold for random graphs $G(n,p)$ at $p = Theta(n^{-2/3})$. Structurally, we characterize minimal non-triangle-covered graphs. Algorithmically, we design exact algorithms for trees and chordal graphs, and present an $(ln n + 1)$-approximation algorithm for general graphs. Collectively, our work unifies complexity-theoretic, structural, and algorithmic analyses of triangle covering, thereby filling a fundamental theoretical gap in this line of research.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into a triangle-covered graph (one in which every vertex belongs to at least one triangle). We first present tight lower bounds on the number of edges in any connected triangle-covered graph of order $n$, and then we characterize all connected graphs that attain this minimum edge count. For a graph $G$, we define the notion of a $Δ$-completion set as a set of non-edges of $G$ whose addition to $G$ results in a triangle-covered graph. We prove that the decision problem of finding a $Δ$-completion set of size at most $tgeq0$ is $mathbb{NP}$-complete and does not admit a constant-factor approximation algorithm under standard complexity assumptions. Moreover, we show that this problem remains $mathbb{NP}$-complete even when the input is restricted to connected bipartite graphs. We then study the problem from an algorithmic perspective, providing tight bounds on the minimum $Δ$-completion set size for several graph classes, including trees, chordal graphs, and cactus graphs. Furthermore, we show that the triangle-covered problem admits an $(ln n +1)$-approximation algorithm for general graphs. For trees and chordal graphs, we design algorithms that compute minimum $Δ$-completion sets. Finally, we show that the threshold for a random graph $mathbb{G}(n, p)$ to be triangle-covered occurs at $n^{-2/3}$.

Problem

• Transforming graphs into triangle-covered graphs via edge modification.
• Determining NP-completeness of finding minimum triangle-completion sets.
• Analyzing triangle-covered graphs across various graph classes and random graphs.

Innovation

• Introduces Delta-completion sets for triangle-covered graphs
• Proves NP-completeness and no constant-factor approximation
• Provides approximation algorithms for various graph classes